I believe this is your question:
"So I had a thought about retirement, and I was wondering how would everyone in the U.S. be able to have a decent retirement fund set up for them. (for low and middle class)"
Your assumptions and the hypotheses are great and I didn't see any problems in it. However, I believe you are overwhelmed by how big is the total amount of the budget they have for the retirement plans.
First of all, money is flowing. There is no bank or any organizations holding such huge amount of money. They do the planning by investment, purchasing bonds, etc. so that they can make sure they have enough amount of money flowing in the market without broken. This is their main concern.
For example, they started the planning and collected the money from 5 people this year. Then, next year they will have 5 other people and so on until they retire. Let's say now they have got the money collected from these people, they will invest the money safely into some organizations and earn the interests. Then, until they retire, they will have the interests to pay them back (and that is why the rules are there to securely ensure the money is flowing without broken, such as "you are not allowed to withdraw a certain amount of money unless it is an emergency")
I feel like I may not describe it clear enough, but my main point is "money is flowing". If you have more questions, feel free to ask. Thanks.